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ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Synchronization and Transient Stability in Power Networks and
Non-Uniform Kuramoto Oscillators Authors: Florian Dorfler and Francesco Bullo
Jin Wei
Submitted in Partial Fulfillment of the Course Requirements for ECEN 689: Cyber Security of the Smart Grid
Instructor: Dr. Deepa Kundur
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Outline
• Introduction and Motivation • Transient Stability Analysis in Power
Networks • Singular Perturbation Analysis and Main
Synchronization Results • Conclusions • My Assessments
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Introduction and Motivation
• The electrical grid is the largest and most complex machine ever made. [2] – Large-scale, complex and rich nonlinear
dynamic behaviors. – Prone to instabilities, which can ultimately lead
to power blackouts. • The blackout happened in 2003 affected a wide swath of
territory in the U.S. and Canada.
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Introduction and Motivation
• Expected developments in Smart Grid – Large number of distributed power sources. – Large-scale heterogeneous networks with stochastic
disturbances. – The detection and rejection of instability mechanisms
will be one of the major challenges.
Transient Stability: the ability of the generators maintain synchronism when subjected to a severe transient disturbance such as a fault on transmission facilities or loss of a large load.
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Introduction and Motivation
• Other existing methods – Treat the transient stability problem as a special case
of the more general synchronization problems, which considers a possibly longer time horizon, drifting generator rotor angles, and local excitation controllers aiming to restore synchronism.
– Do not result in simple conditions to check if a power system synchronizes for a given network state and parameters.
It is an outstanding problem to relate synchronization and transient stability of a power network to the underlying network parameters, state, and topology.
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Transient Stability Analysis in Power Networks
• Previous work – Structure-preserving DAE power network model – Network-reduced ODE power network model
• Network reduced to active nodes (generators) by using kron-reduction technique.
• Admittance matrix Yred induces complete all-to-all coupling graph
Fig 1. Network Reduction Procedure for New England 39-bus test system [3]
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Transient Stability Analysis in Power Networks
• Classic interconnected swing equations
– max power transferred between i & j
– line loss between i & j
– effective power input of i
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Transient Stability Analysis in Power Networks
• Transient stability and synchronization – Frequency equilibrium: ( , )=(0,0) for all i. – Synchronous equilibrium: and 0 for all (i, j) as .
• Classic analysis tools: Hamiltonian arguments
where
hyperbolic type-k equilibium Dimension-reduced gradient flow analysis:
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Transient Stability Analysis in Power Networks
• Classic analysis tools: Hamiltonian arguments – Shortcomings:
• Over simplify the model: e.g. assuming φij=0 • Implement the numerical procedures rather than induce
concise conditions: e.g. only indicating “sufficiently small” transfer conductances without quantifying the smallness.
• Not setup the relationship between the synchronization & transient stability in power networks and the underlying network state, parameters & topology.
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Singular Perturbation Analysis and Main Synchronization Results
Consensus protocol in Rn
• n identical agents with state variable • Objective: state agreement: as
Kuramoto model in Tn
• n non-identical oscillators with phase • Objective phase locking: frequency entrainment: phase synchronization:
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
• Singular perturbation analysis – Time-scale separation in power network model
– Singular perturbation parameter: – Reduced dynamics on slow time-scale (for ϵ<<1)
– Assume the non-uniform Kuramoto model synchronizes exponentially, , ϵ*>0, s.t. ϵ<ϵ* and : θi(t)power network - θi(t)non-uniform kuramoto=O(ϵ). For ϵ and the network losses φij sufficiently small, O(ϵ) converges to 0 asymptotically.
Singular Perturbation Analysis
non-uniform Kuramoto model
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
• Discussion of the assumption ϵ is sufficiently small. – Generator internal control effects (e.g. local excitation
controllers) imply – Topological equivalence independent of ϵ: 1st-order and
2nd-order models have the same equilibria, the Jacobians have the same inertia, and the regions of attractions are bounded by the same separatrices.
– simulation studies show accurate approximation even for large ϵ.
– Non-uniform Kuramoto corresponds to reduced gradient system used successfully in academia and industry since 1978.
Singular Perturbation Analysis
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Synchronization of Non-Uniform Kuramoto Oscillators
• Non-uniformity in network: Di, ωi, Pij, φij
• Phase shift φij induces lossless and lossy coupling
• Synchronization analysis
– Phase locking: – Frequency entrainment: – Phase synchronization:
non-uniform Kuramoto model
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Synchronization of Non-Uniform Kuramoto Oscillators
• Synchronization condition:
Worst lossless coupling
Worst non-uniformity
Worst lossy coupling
The network connectivity has to dominate the network’s non-uniformity, the network’s losses, and the lack of phase locking
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Synchronization of Non-Uniform Kuramoto Oscillators
• Synchronization condition:
• Phase Locking
• Frequency entrainment – From all initial conditions in a arc, exponential
frequency synchronization. –
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
• Main proof ideas – An analogous result guarantees synchronization of
the non-uniform Kuramoto oscillators whenever the graph induced by P has a globally reachable node.
frequency entrainment in consensus protocol in Rn
Synchronization of Non-Uniform Kuramoto Oscillators
where
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Synchronization of Non-Uniform Kuramoto Oscillators
• Alternative synchronization condition – Non-uniform Kuramoto model:
– Condition:
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
Conclusions
• Study the synchronization and transient stability problem for a network-reduction model of a power system.
• Provide the conditions depending on network parameters and initial phase differences suffice for the synchronization of non-uniform Kuramoto oscillators as well as the transient stability of the power network model.
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
My Assessments
• Suggestions on future works – Specify the synchronization frequency. – Specify the exponential rate of the frequency
synchronization. – Develop the stochastic analysis instead of the
worst-case analysis. – Study the more general power network (e.g.
the underlying graph is not complete). – Provide the simulations on practical power
test system.
ECEN 689: Cyber Security of the Smart Grid, Spring 2011 Class Presentation Jin Wei
References
1. F. Dorfler and F. Bullo, “Transient stability analysis in power networks and synchronization of non-uniform Kuramoto oscillators,” in ACC, June 2010, pp. 930-937.
2. P. F. Schewe, The Grid: A Journey Through the Heart of Our Electrified World, Joseph Henry Press, 2006.
3. F. Dorfler and F. Bullo, “Spectral Analysis of Synchronization in a Lossless Structure-Preserving Power Network Model,” in Proceedings of the First IEEE International Conference on Smart Grid Communications, 2010, pp. 179-184.